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A relation between one-point and multi-point Seshadri constants
Journal of Algebra 274 (2004) 643–651 www.elsevier.com/locate/jalgebra
A relation between one-point and multi-point Seshadri constants Joaquim Roé 1 Departament de Matemàtiques, Universitat Autonòma de Barcelona, 08193 Bellaterra (Barcelona), Spain Received 25 February 2003 Communicated by Craig Huneke
Abstract T. Szemberg proposed in 2001 a generalization to arbitrary varieties of M. Nagata’s 1959 open conjecture, which claims that the Seshadri constant of r 9 very general points of the projective plane is maximal. Here we prove that Nagata’s original conjecture implies Szemberg’s for all smooth surfaces X with an ample divisor L generating NS(X) and such that L2 is a square. More generally, we prove the inequality εn−1 (L, r) εn−1 (L, 1)εn−1 OPn (1), r , where εn−1 (L, r) stands for the (n − 1)-dimensional Seshadri constant of the ample divisor L at r very general points of a normal projective variety X, and n = dim X. 2004 Elsevier Inc. All rights reserved. Keywords: Seshadri constant; Nagata’s conjecture
1. Introduction Let X be a normal projective variety of dimension n over an algebraically closed field k, and L an ample divisor. Given r points p1 , . . . , pr ∈ X and an integer 1 d n, the d-dimensional Seshadri constant of L at the points p1 , . . . , pr is the real number E-mail address: [email protected]. 1 Partially supported by 2000SGR-00028 (Catalonia), MCyT BFM2002-012040 (Spain), and EAGER
(European Union). 0021-8693/$ – see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.jalgebra.2003.10.009
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εd (L, p1 , . . . , pr ) =
d
Ld · Z , inf Z multpi Z
where Z runs over all positive d-dimensional cycle s (see [1, 1.1], or [2] for the original definition). As L is ample, we have Ld · Z > 0 for all Z, so (Ld · Z)/( multpi Z) ∈ R+ ∪ ∞. Moreover, there exist Z which contain some point pi , and therefore the Seshadri constant is indeed a finite real number. Most work on Seshadri constants deals with the d = 1 case, and usually one writes ε(L, p1 , . . . , pr ) = ε1 (L, p1 , . . . , pr ); we shall be concerned here with the codimension 1 case (d = n − 1). Also, we use the shorthand notation εd (L, r) = εd (L, p1 , . . . , pr ) for very general points p1 , . . . , pr (i.e., in the intersection of countably many Zariski open subsets of Xr ) which is the case we are mostly interested in. In connection with his solution to the fourteenth problem of Hilbert, Nagata posed in [3] (in different terminology) the following conjecture concerning Seshadri constants of the plane: Conjecture 1 (Nagata). If r 9, then √ ε OP 2 (1), r = 1/ r. If r = s 2 is a square, then it is not hard to prove that the conjecture is true, and in fact the point of Nagata’s stronger result for the r = s 2 > 9 case is that the infimum appearing in the definition of the Seshadri constant is not achieved by any plane curve. In a variety of dimension n it is also not difficult to prove that ε(L, p1 , . . . , pr )
n
Ln r
for every set of r points (see [4, Remark 1], for example), so Nagata’s conjecture claims that the Seshadri constant of a very general set of r 9 points in the plane is maximal. All available information on Seshadri constants (see [5–11] for the case of surfaces, [12–14] for dimension n > 2) suggests that, in fact, in an arbitrary variety, for r large enough, the Seshadri constant of r very general points is maximal. This led Szemberg to propose in [1] the following generalization: Conjecture 2 (Nagata–Szemberg). Given a smooth variety of dimension n and L an ample divisor on X, there exists a number r0 = r0 (X, L) such that for every r r0 ε(L, r) =
n
Ln . r
This note is devoted to the following result, that gives a lower bound for (n − 1)-dimensional Seshadri constants of r very general points in a variety, relating them to the analogous constants of one point in the same variety and of r points in projective space:
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Theorem 3. Let X be a normal projective variety of dimension n 2, L an ample divisor. Then for every smooth point p ∈ X and r 1, εn−1 (L, r) εn−1 (L, p)εn−1 OPn (1), r . Combining it with known results on the value of the Seshadri constants in projective space, Theorem 3 implies more explicit relations between r-point and 1-point Seshadri constants. The consequences support the Nagata–Szemberg conjecture, especially in the case of surfaces (note that for surfaces, (n − 1)-dimensional Seshadri constants are the usual Seshadri constants). Corollary 4. Suppose r = s n for some integer s. Then for every smooth point p ∈ X and r 1, εn−1 (L, r) εn−1 (L, p)/s. Proof. Use G.V. Choodnovsky’s result [13] that εn−1 (OPn (1), s n ) = 1/s.
✷
If r is not the nth power of an integer, then we do not know the exact value of εn−1 (OPn (1), r) (there is a conjecture similar to Nagata’s, posed by Choodnovsky in the same paper [13], and by A. Iarrobino in [15]). However, B. Harbourne pointed out that, using results of J. Alexander and A. Hirschowitz and of M. Hochster and C. Huneke, one can prove an asymptotically optimal bound for εn−1 (OPn (1), r). Combining it with Theorem 3, in Section 2 we prove the following asymptotic bound for εn−1 (L, r) that depends only on the (n − 1)-dimensional Seshadri constant of L at a smooth point p ∈ X: Corollary 5. For every ε > 0 there exists an integer s, depending only on ε and n, such that for every smooth point p ∈ X and r s, √ εn−1 (L, r) n r εn−1 (L, p) − ε. Corollary 6. If X is a normal projective surface, L an ample divisor on X, then for every smooth point p ∈ X and r 9, Nagata’s conjecture implies that ε(L, r)
ε(L, p) √ . r
Observe that this tells us that Nagata’s conjecture (on the plane) implies the Nagata– Szemberg√conjecture on a large family of surfaces. Indeed, the obtained bound is equal to ε(L, p)/ L2 times the conjectured value of ε(L, r), so we have the following: Corollary 7. If X is a normal projective √ surface, L an ample divisor on X and p ∈ X is a smooth point such that ε(L, p) = L2 , then Nagata’s conjecture implies the Nagata– Szemberg conjecture on (X, L) with r0 (X, L) 9.
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In particular, we can apply this to complex surfaces with Picard number 1, using A. Steffens’ result √[4, Proposition 1], which says that if L is an ample generator of NS(X) then ε(L, p) L2 for very general points: Corollary 8. Let X be a smooth projective surface defined over C, L an ample generator of NS(X) and assume L2 = d 2 is a square. Then Nagata’s conjecture implies the Nagata– Szemberg conjecture on X with r0 (X, L) 9. In particular, the Seshadri constant of r 9 very general points on a complex surface X with Picard number equal to 1, is maximal if both L2 and the number of points r are squares. This can be compared to B. Harbourne’s result [11, I.1] (over an arbitrary base field and with no assumption on the Picard number) that the Seshadri constant is maximal whenever L is very ample, rL2 is a square, and r L2 . Also, known bounds approximating Nagata’s conjecture give new bounds on surfaces; √ for instance, H. Tutaj-Gasi´nska’s bound in [16] showing that ε(OP2C (1), r) 1/( r + 1/12) gives the following: Corollary 9. If X is a normal projective surface defined over C, then for every smooth point p ∈ X and r > 9, ε(L, p) ε(L, r) . 1 r + 12 In a similar vein, Harbourne’s bounds on Seshadri constants of P2 in [17] and [11] imply that Corollary 10. Let X be a normal projective surface, p ∈ X a smooth point and r 1. Then for every pair of integers 1 s r, 1 d, it holds ε(L, r)
s rd ε(L, p), d s ε(L, p),
if s 2 rd 2 , if s 2 rd 2 .
It should be mentioned, however, that both Corollaries 9 and 10 are usually weaker (but stronger for some surfaces and numbers of points) than Harbourne’s results on algebraic surfaces of [11], where he gets the bounds ε(L, r)
s rd , dL2 s ,
if s 2 rd 2 L2 , if s 2 rd 2 L2
for very ample L, assuming moreover that r L2 . The proof of Theorem 3 is based on the idea, due to L. Évain (see [18]) that r-point Seshadri constants of the plane can be computed by means of homothetic collisions of fat points. For convenience of the exposition we express this, generalized to n-dimensional
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projective space, in terms of the order of a nonreduced curve singularity, rather than collisions. Then, we observe that it is enough to know the formal germ of such a singularity, and the fact that the completion of the local ring at a smooth point of a variety is a ring of formal power series that only depends on the dimension of the variety, to obtain the bound. It might be interesting to note that the Viro method developed by E. Shustin in [19] can also be used to relate the existence of singular curves on P2 with the existence of singular curves in algebraic surfaces, see for instance [20, §5] or [8, 3.A].
2. Singularities of arrangements of multiple lines To lighten notations for rings of polynomials and of power series, we write x = (x0 , . . . , xn ) for a collection of variables, so k[x] and k[[x]] denote k[x0 , . . . , xn ] and k[[x0, . . . , xn ]], ˆ will be the maximal ideals generated by x0 , . . . , xn in k[x] and respectively. Also, m and m k[[x]], respectively, and for every point p = [ξ0 : · · · : ξn ] in projective n-space, we use the notation Ip (respectively Iˆp ) for the ideal generated by the 2 × 2 minors of the matrix
ξ0 x0
. . . ξn . . . xn
in k[x] (respectively in k[[x]]). Given distinct points p1 , . . . , pr ∈ Pn and m = (m1 , . . . , mr ) ∈ Zr0 , we define αm (p1 , . . . , pr ) to be the minimal degree of a homogeneous polynomial vanishing to order mi at pi . As Ip is homogeneous for all p, this number coincides with the maximal integer α such that I=
r
Ipmi i ⊂ mα ,
i=1
ˆ α . In other words, αm (p1 , . . . , pr ) is the order at or equivalently, such that Iˆ = Iˆpmi i ⊂ m n+1 the origin of A of the arrangement of multiple lines defined by I (which is the affine cone over the fat point scheme consisting of the points pi with multiplicities mi ). Remark 11. The definition of α and the (n − 1)-dimensional Seshadri constants immediately give that ∀m, r n−1 αm (p1 , . . . , pr ) εn−1 OPn (1), p1 , . . . , pr mi . i=1
Let X be a variety of dimension n 2, q ∈ X a smooth point, and fix uniformizing parameters x1 , . . . , xn in some neighborhood V centered at q. So (see [21, §III.6], for example) the germs of x1 , . . . , xn at q generate the maximal ideal in the local ring OX,q and the morphism of k-algebras k[x] → OX (V ) determines an étale morphism X,q . ϕ : V → An ∼ = Tq X (where V ⊂ X is open) and an isomorphism k[[x]] → O
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To every p = [ξ0 : · · · : ξn ] ∈ Pn such that ξ0 = 0 and p = [1 : 0 : · · · : 0] (so that p = [ξ1 : · · · : ξn ] ∈ Pn−1 ), we shall assign an irreducible curve Cp , smooth at q, and a regular parameter x¯p ∈ OCp ,q . Define Cp as the closure of the component through q of ϕ −1 (Lp ), where Lp ⊂ An is the affine cone over p , i.e., the line through the origin in the direction determined by p . As ϕ is étale, Cp is smooth at q, and the ideal ICp ⊂ OX,q of its germ at q is generated by the 2 × 2 minors of the matrix
ξ1 . . . ξn . x1 . . . xn Now consider xp = xi ξ0 /ξi for some ξi = 0, i 1. It is easy to see that the restriction x¯p of xp to Cp does not depend on the choice of i, and that it is a uniformizing parameter for the curve. Also, abusing notation, in OCp ×X,(q,q) = OCp ,q ⊗ OX,q we write xp = x¯p ⊗ 1, x1 = 1 ⊗ x1 , . . . , and xn = 1 ⊗ xn ; then the ideal of the germ of the diagonal ∆(Cp ) ⊂ Cp × X is generated by the 2 × 2 minors of the matrix
ξ0 xp
ξ1 x1
. . . ξn , . . . xn
in other words, ∆(Cp ) is the closure of the component through the origin of ψ −1 (Lp ), where Lp is the affine cone over p ∈ Pn , i.e., a line Lp ⊂ An+1 = T(q,q) (Cp × X), and ψ is the étale morphism given by the parameters xp , x1 , . . . , xn . With these notations, Theorem 3 follows from the more precise proposition: Proposition 12. Let X be a variety of dimension n 2, L an ample divisor, q ∈ X a smooth point, p1 , . . . , pr ∈ Pn \ [1 : 0 : · · · : 0] distinct points not on the hyperplane ξ0 = 0. Then for very general points qk ∈ Cpk we have εn−1 (L, q1 , . . . , qr ) εn−1 (L, q)εn−1 OPn (1), p1 , . . . , pr . Proof. First note that due to the semicontinuity of multiplicity (see [22, §8] or [23, §3]), for each component H of the Hilbert scheme of hypersurfaces in X, and each system of multiplicities m, the sets of points (q1 , . . . , qr ) such that there is Y ∈ H with multiplicity mi at qi form a Zariski-closed subset of Xr . Thus, it will be enough to prove that, given H and m, the existence of Y ∈ H with multiplicity mk at qk for general qk ∈ Cpk implies r n−1 mk . Y · Ln−1 εn−1 (L, q)εn−1 OPn (1), p1 , . . . , pr k=1
So, fix H and m, and assume that for general points qk ∈ Cpk there is a hypersurface Y ∈ H going through qk with multiplicity at least mk . In the local ring of Cpk at ∆(q) = (q, . . . , q), the 1 ⊗ · · ·⊗ x¯pk ⊗ · · · ⊗ 1, k = 1, . . . , r, form a regular system of parameters; abusing notation we call them simply xpk . Let Γ ⊂ Cpk be the irreducible curve defined locally by the equations xp1 = xpi , i = 2, . . . , r, which is obviously smooth at ∆(q), and admits x0 := x¯ p1 ∈ OΓ,∆(q) as a local parameter.
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For every pk = [ξk,0 : · · · : ξk,n ], the ideal Jpk ⊂ OΓ ×X,∆(q) generated by the 2 × 2 minors of the matrix
ξk,0 . . . ξk,n , x0 . . . xn defines the germ of a curve Cp k (the preimage of a line in An+1 ∼ = T∆(q) (Γ × X)) whose projection to X is exactly Cpk ; more precisely, the fiber of Cp k over γ = (q1 , . . . , qn ) ∈ Γ is qk ∈ Cpk . Consider now the diagonals ∆i,j = {(q1 , . . . , qr ) ∈ Cpk | qi = qj }, ∆ = ∆i,j , and U = Γ \ ∆. As the points p1 , . . . , pr are distinct, U is a nonempty open subset of Γ . The assumption on the existence of Y tells us that there is an effective Weil divisor YU ⊂ U × X flat over U whose fiber over (q1 , . . . , qr ) ∈ U belongs to H and has multiplicity at least mi at qi . By the smoothness of Γ at ∆(q), YU can be extended to a flat family Y ⊂ (U ∪ {∆(q)}) × X, and then the condition on the multiplicity of the fibers of Y means that Y contains the arrangement of multiple curves whose germ at ∆(q) is defined by the ideal J=
r
Jpmkk ,
k=1
and this implies that the fiber Yq of Y over ∆(q) has multiplicity at least equal to the order of this arrangement at ∆(q), i.e., at least equal to the maximal integer α such that J ⊂ (x0 , . . . , xn )α . This can be computed equivalently as the order of the completion Jˆ A1 ×X,(0,q) ∼ in O = k[[x]] but, by construction of the curves Cp k , one has Jˆ = Iˆ as defined above, so this order is exactly αm (p1 , . . . , pr ). We remark also that, by the smoothness of X in a neighborhood of q, Y is defined by a principal ideal at q, so we get a Weil divisor Yq ⊆ Yq with Ln−1 · Yq = Ln−1 · Y, multq Yq αm (p1 , . . . , pr ), which together with the definition of the 1-point Seshadri constant, gives Ln−1 · Y εn−1 (X, q)αm (p1 , . . . , pr ), and then it is enough to apply the bound of Remark 11. ✷ Proof of Corollary 5. By Theorem 3 it is enough to see that √ lim εn−1 OPn (1), r n r 1 r→∞
(which in fact means that one has an equality, the converse inequality being well-known). More precisely, we shall prove that given k > 0 there exists sk = sk (n) such that if r sk then for all m = (m1 , . . . , mr ) and general points p1 , . . . , pr , mi k + 1 . αm (p1 , . . . , pr ) √ · n n−1 k+n r
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So let F be a homogeneous polynomial defining a hypersurface of degree d in Pn which has multiplicity mi at pi for general points p1 , . . . , pr . Then, by the genericity of the points, for every permutation σ ∈ Sr there is a polynomial Fσ which has multiplicity mi at the point pσ (i) . Therefore G = σ ∈Sr Fσ is a polynomial of degree D = r! d which has (the same) multiplicity M = (r − 1)! mi at p1 , . . . , pr , and Gk+n has degree (k + n)D and multiplicity (k + n)M at each point. By [24, Theorem 1.1(a)], applied to the ideal I of the (reduced) scheme {p1 , . . . , pr }, this implies the existence of hypersurfaces of degree t (k + n)D/M with multiplicity at least k + 1 at p1 , . . . , pr . Now write m = (k + 1, . . . , k + 1); by√[12, Corollary 1.2], there is sk (n) such that if r sk (n) then αm (p1 , . . . , pr ) (k +√1) n r (again, because the points are general). Therefore we get (k + n)D/M (k + 1) n r and 1 D k+1 d √ , = · n n−1 rM k +n mi r as desired. ✷
Acknowledgments We thank B. Harbourne and S. Kleiman for many valuable comments, which largely improved the paper.
References [1] T. Szemberg, Global and local positivity of line bundles, Habilitation, 2001, 42 pp. [2] J. Demailly, Singular Hermitian metrics on positive line bundles, in: K. Hulek, et al. (Eds.), Complex Algebraic Varieties (Bayreuth, 1990), in: Lecture Notes in Math., vol. 1507, Springer, 1992, pp. 87–104. [3] M. Nagata, On the fourteenth problem of Hilbert, Amer. J. Math. 81 (1959) 766–772. [4] A. Steffens, Remarks on Seshadri constants, Math. Z. 227 (3) (1998) 505–510. [5] G. Xu, Ample line bundles on smooth surfaces, J. Reine Angew. Math. 469 (1995) 199–209. [6] R. Lazarsfeld, Lectures on linear series, in: J. Kollár (Ed.), Complex Algebraic Geometry (Park City, UT, 1993), no. 3, in: IAS/Park City Math. Series, Amer. Math. Soc., Providence, RI, 1997, pp. 161–219, with the assistance of Guillermo Fernández del Busto. [7] O. Küchle, Multiple point Seshadri constants and the dimension of adjoint linear series, Ann. Inst. Fourier 46 (1) (1996) 63–71. [8] P. Biran, Constructing new ample divisors out of old ones, J. Reine Angew. Math. 98 (1) (1998) 113–136. [9] T. Bauer, Seshadri constants on algebraic surfaces, Math. Ann. 313 (3) (1999) 547–583. [10] T. Szemberg, H. Tutaj-Gasi´nska, General blow-ups of the projective plane, Proc. Amer. Math. Soc. 130 (9) (2002) 2515–2524. [11] B. Harbourne, Seshadri constants and very ample divisors on algebraic surfaces, J. Reine Angew. Math. 559 (2003) 115–122. [12] J. Alexander, A. Hirschowitz, An asymptotic vanishing theorem for generic unions of multiple points, Invent. Math. 140 (2000) 303–325. [13] G.V. Choodnovsky, Sur la construction de Rees et Nagata pour le 14e problème de Hilbert, C. R. Acad. Sci. Paris 286 (23) (1978) 1133–1135. [14] J.-M. Hwang, J.-H. Keum, Seshadri-exceptional foliations, Math. Ann. 325 (2) (2003) 287–297.
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[15] A. Iarrobino, Inverse system of a symbolic power, III: Thin algebras and fat points, Compositio Math. 108 (3) (1997) 319–356. [16] H. Tutaj-Gasi´nska, A bound for Seshadri constants on P2 , Math. Nachr. 257 (2003) 108–116. [17] B. Harbourne, On Nagata’s conjecture, J. Algebra 236 (2001) 692–702. [18] L. Évain, Calculs des dimensions de systèmes linéaires de courbes planes par collisions de gros points, C. R. Acad. Sci. Paris 325 (I) (1997) 1305–1308. [19] E. Shustin, Gluing of singular and critical points, Topology 37 (1998) 195–217. [20] T. Keilen, I. Tyomkin, Existence of curves with prescribed topological singularities, Trans. Amer. Math. Soc. 354 (5) (2002) 1827–1860. [21] D. Mumford, The Red Book of Varieties and Schemes, expanded edition, includes the Michigan Lectures (1974) on Curves and Their Jacobians with contributions by Enrico Arbarello, in: Lecture Notes in Math., vol. 1358, Springer-Verlag, Berlin, 1979. [22] M. Lejeune-Jalabert, B. Teissier, Contribution à l’étude des singularités du point de vue du polygone de Newton, PhD thesis, Université de Paris VII, Paris, 1973. [23] J. Lipman, Equimultiplicity, reduction, and blowing up, in: Commutative Algebra (Fairfax, VA, 1979), in: Lecture Notes in Pure and Appl. Math., vol. 68, Dekker, New York, 1982, pp. 111–147. [24] M. Hochster, C. Huneke, Comparison of symbolic and ordinary powers of ideals, Invent. Math. 147 (2) (2002) 349–369.
A relation between one-point and multi-point Seshadri constants Joaquim Roé 1 Departament de Matemàtiques, Universitat Autonòma de Barcelona, 08193 Bellaterra (Barcelona), Spain Received 25 February 2003 Communicated by Craig Huneke
Abstract T. Szemberg proposed in 2001 a generalization to arbitrary varieties of M. Nagata’s 1959 open conjecture, which claims that the Seshadri constant of r 9 very general points of the projective plane is maximal. Here we prove that Nagata’s original conjecture implies Szemberg’s for all smooth surfaces X with an ample divisor L generating NS(X) and such that L2 is a square. More generally, we prove the inequality εn−1 (L, r) εn−1 (L, 1)εn−1 OPn (1), r , where εn−1 (L, r) stands for the (n − 1)-dimensional Seshadri constant of the ample divisor L at r very general points of a normal projective variety X, and n = dim X. 2004 Elsevier Inc. All rights reserved. Keywords: Seshadri constant; Nagata’s conjecture
1. Introduction Let X be a normal projective variety of dimension n over an algebraically closed field k, and L an ample divisor. Given r points p1 , . . . , pr ∈ X and an integer 1 d n, the d-dimensional Seshadri constant of L at the points p1 , . . . , pr is the real number E-mail address: [email protected]. 1 Partially supported by 2000SGR-00028 (Catalonia), MCyT BFM2002-012040 (Spain), and EAGER
(European Union). 0021-8693/$ – see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.jalgebra.2003.10.009
644
J. Roé / Journal of Algebra 274 (2004) 643–651
εd (L, p1 , . . . , pr ) =
d
Ld · Z , inf Z multpi Z
where Z runs over all positive d-dimensional cycle s (see [1, 1.1], or [2] for the original definition). As L is ample, we have Ld · Z > 0 for all Z, so (Ld · Z)/( multpi Z) ∈ R+ ∪ ∞. Moreover, there exist Z which contain some point pi , and therefore the Seshadri constant is indeed a finite real number. Most work on Seshadri constants deals with the d = 1 case, and usually one writes ε(L, p1 , . . . , pr ) = ε1 (L, p1 , . . . , pr ); we shall be concerned here with the codimension 1 case (d = n − 1). Also, we use the shorthand notation εd (L, r) = εd (L, p1 , . . . , pr ) for very general points p1 , . . . , pr (i.e., in the intersection of countably many Zariski open subsets of Xr ) which is the case we are mostly interested in. In connection with his solution to the fourteenth problem of Hilbert, Nagata posed in [3] (in different terminology) the following conjecture concerning Seshadri constants of the plane: Conjecture 1 (Nagata). If r 9, then √ ε OP 2 (1), r = 1/ r. If r = s 2 is a square, then it is not hard to prove that the conjecture is true, and in fact the point of Nagata’s stronger result for the r = s 2 > 9 case is that the infimum appearing in the definition of the Seshadri constant is not achieved by any plane curve. In a variety of dimension n it is also not difficult to prove that ε(L, p1 , . . . , pr )
n
Ln r
for every set of r points (see [4, Remark 1], for example), so Nagata’s conjecture claims that the Seshadri constant of a very general set of r 9 points in the plane is maximal. All available information on Seshadri constants (see [5–11] for the case of surfaces, [12–14] for dimension n > 2) suggests that, in fact, in an arbitrary variety, for r large enough, the Seshadri constant of r very general points is maximal. This led Szemberg to propose in [1] the following generalization: Conjecture 2 (Nagata–Szemberg). Given a smooth variety of dimension n and L an ample divisor on X, there exists a number r0 = r0 (X, L) such that for every r r0 ε(L, r) =
n
Ln . r
This note is devoted to the following result, that gives a lower bound for (n − 1)-dimensional Seshadri constants of r very general points in a variety, relating them to the analogous constants of one point in the same variety and of r points in projective space:
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Theorem 3. Let X be a normal projective variety of dimension n 2, L an ample divisor. Then for every smooth point p ∈ X and r 1, εn−1 (L, r) εn−1 (L, p)εn−1 OPn (1), r . Combining it with known results on the value of the Seshadri constants in projective space, Theorem 3 implies more explicit relations between r-point and 1-point Seshadri constants. The consequences support the Nagata–Szemberg conjecture, especially in the case of surfaces (note that for surfaces, (n − 1)-dimensional Seshadri constants are the usual Seshadri constants). Corollary 4. Suppose r = s n for some integer s. Then for every smooth point p ∈ X and r 1, εn−1 (L, r) εn−1 (L, p)/s. Proof. Use G.V. Choodnovsky’s result [13] that εn−1 (OPn (1), s n ) = 1/s.
✷
If r is not the nth power of an integer, then we do not know the exact value of εn−1 (OPn (1), r) (there is a conjecture similar to Nagata’s, posed by Choodnovsky in the same paper [13], and by A. Iarrobino in [15]). However, B. Harbourne pointed out that, using results of J. Alexander and A. Hirschowitz and of M. Hochster and C. Huneke, one can prove an asymptotically optimal bound for εn−1 (OPn (1), r). Combining it with Theorem 3, in Section 2 we prove the following asymptotic bound for εn−1 (L, r) that depends only on the (n − 1)-dimensional Seshadri constant of L at a smooth point p ∈ X: Corollary 5. For every ε > 0 there exists an integer s, depending only on ε and n, such that for every smooth point p ∈ X and r s, √ εn−1 (L, r) n r εn−1 (L, p) − ε. Corollary 6. If X is a normal projective surface, L an ample divisor on X, then for every smooth point p ∈ X and r 9, Nagata’s conjecture implies that ε(L, r)
ε(L, p) √ . r
Observe that this tells us that Nagata’s conjecture (on the plane) implies the Nagata– Szemberg√conjecture on a large family of surfaces. Indeed, the obtained bound is equal to ε(L, p)/ L2 times the conjectured value of ε(L, r), so we have the following: Corollary 7. If X is a normal projective √ surface, L an ample divisor on X and p ∈ X is a smooth point such that ε(L, p) = L2 , then Nagata’s conjecture implies the Nagata– Szemberg conjecture on (X, L) with r0 (X, L) 9.
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In particular, we can apply this to complex surfaces with Picard number 1, using A. Steffens’ result √[4, Proposition 1], which says that if L is an ample generator of NS(X) then ε(L, p) L2 for very general points: Corollary 8. Let X be a smooth projective surface defined over C, L an ample generator of NS(X) and assume L2 = d 2 is a square. Then Nagata’s conjecture implies the Nagata– Szemberg conjecture on X with r0 (X, L) 9. In particular, the Seshadri constant of r 9 very general points on a complex surface X with Picard number equal to 1, is maximal if both L2 and the number of points r are squares. This can be compared to B. Harbourne’s result [11, I.1] (over an arbitrary base field and with no assumption on the Picard number) that the Seshadri constant is maximal whenever L is very ample, rL2 is a square, and r L2 . Also, known bounds approximating Nagata’s conjecture give new bounds on surfaces; √ for instance, H. Tutaj-Gasi´nska’s bound in [16] showing that ε(OP2C (1), r) 1/( r + 1/12) gives the following: Corollary 9. If X is a normal projective surface defined over C, then for every smooth point p ∈ X and r > 9, ε(L, p) ε(L, r) . 1 r + 12 In a similar vein, Harbourne’s bounds on Seshadri constants of P2 in [17] and [11] imply that Corollary 10. Let X be a normal projective surface, p ∈ X a smooth point and r 1. Then for every pair of integers 1 s r, 1 d, it holds ε(L, r)
s rd ε(L, p), d s ε(L, p),
if s 2 rd 2 , if s 2 rd 2 .
It should be mentioned, however, that both Corollaries 9 and 10 are usually weaker (but stronger for some surfaces and numbers of points) than Harbourne’s results on algebraic surfaces of [11], where he gets the bounds ε(L, r)
s rd , dL2 s ,
if s 2 rd 2 L2 , if s 2 rd 2 L2
for very ample L, assuming moreover that r L2 . The proof of Theorem 3 is based on the idea, due to L. Évain (see [18]) that r-point Seshadri constants of the plane can be computed by means of homothetic collisions of fat points. For convenience of the exposition we express this, generalized to n-dimensional
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projective space, in terms of the order of a nonreduced curve singularity, rather than collisions. Then, we observe that it is enough to know the formal germ of such a singularity, and the fact that the completion of the local ring at a smooth point of a variety is a ring of formal power series that only depends on the dimension of the variety, to obtain the bound. It might be interesting to note that the Viro method developed by E. Shustin in [19] can also be used to relate the existence of singular curves on P2 with the existence of singular curves in algebraic surfaces, see for instance [20, §5] or [8, 3.A].
2. Singularities of arrangements of multiple lines To lighten notations for rings of polynomials and of power series, we write x = (x0 , . . . , xn ) for a collection of variables, so k[x] and k[[x]] denote k[x0 , . . . , xn ] and k[[x0, . . . , xn ]], ˆ will be the maximal ideals generated by x0 , . . . , xn in k[x] and respectively. Also, m and m k[[x]], respectively, and for every point p = [ξ0 : · · · : ξn ] in projective n-space, we use the notation Ip (respectively Iˆp ) for the ideal generated by the 2 × 2 minors of the matrix
ξ0 x0
. . . ξn . . . xn
in k[x] (respectively in k[[x]]). Given distinct points p1 , . . . , pr ∈ Pn and m = (m1 , . . . , mr ) ∈ Zr0 , we define αm (p1 , . . . , pr ) to be the minimal degree of a homogeneous polynomial vanishing to order mi at pi . As Ip is homogeneous for all p, this number coincides with the maximal integer α such that I=
r
Ipmi i ⊂ mα ,
i=1
ˆ α . In other words, αm (p1 , . . . , pr ) is the order at or equivalently, such that Iˆ = Iˆpmi i ⊂ m n+1 the origin of A of the arrangement of multiple lines defined by I (which is the affine cone over the fat point scheme consisting of the points pi with multiplicities mi ). Remark 11. The definition of α and the (n − 1)-dimensional Seshadri constants immediately give that ∀m, r n−1 αm (p1 , . . . , pr ) εn−1 OPn (1), p1 , . . . , pr mi . i=1
Let X be a variety of dimension n 2, q ∈ X a smooth point, and fix uniformizing parameters x1 , . . . , xn in some neighborhood V centered at q. So (see [21, §III.6], for example) the germs of x1 , . . . , xn at q generate the maximal ideal in the local ring OX,q and the morphism of k-algebras k[x] → OX (V ) determines an étale morphism X,q . ϕ : V → An ∼ = Tq X (where V ⊂ X is open) and an isomorphism k[[x]] → O
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To every p = [ξ0 : · · · : ξn ] ∈ Pn such that ξ0 = 0 and p = [1 : 0 : · · · : 0] (so that p = [ξ1 : · · · : ξn ] ∈ Pn−1 ), we shall assign an irreducible curve Cp , smooth at q, and a regular parameter x¯p ∈ OCp ,q . Define Cp as the closure of the component through q of ϕ −1 (Lp ), where Lp ⊂ An is the affine cone over p , i.e., the line through the origin in the direction determined by p . As ϕ is étale, Cp is smooth at q, and the ideal ICp ⊂ OX,q of its germ at q is generated by the 2 × 2 minors of the matrix
ξ1 . . . ξn . x1 . . . xn Now consider xp = xi ξ0 /ξi for some ξi = 0, i 1. It is easy to see that the restriction x¯p of xp to Cp does not depend on the choice of i, and that it is a uniformizing parameter for the curve. Also, abusing notation, in OCp ×X,(q,q) = OCp ,q ⊗ OX,q we write xp = x¯p ⊗ 1, x1 = 1 ⊗ x1 , . . . , and xn = 1 ⊗ xn ; then the ideal of the germ of the diagonal ∆(Cp ) ⊂ Cp × X is generated by the 2 × 2 minors of the matrix
ξ0 xp
ξ1 x1
. . . ξn , . . . xn
in other words, ∆(Cp ) is the closure of the component through the origin of ψ −1 (Lp ), where Lp is the affine cone over p ∈ Pn , i.e., a line Lp ⊂ An+1 = T(q,q) (Cp × X), and ψ is the étale morphism given by the parameters xp , x1 , . . . , xn . With these notations, Theorem 3 follows from the more precise proposition: Proposition 12. Let X be a variety of dimension n 2, L an ample divisor, q ∈ X a smooth point, p1 , . . . , pr ∈ Pn \ [1 : 0 : · · · : 0] distinct points not on the hyperplane ξ0 = 0. Then for very general points qk ∈ Cpk we have εn−1 (L, q1 , . . . , qr ) εn−1 (L, q)εn−1 OPn (1), p1 , . . . , pr . Proof. First note that due to the semicontinuity of multiplicity (see [22, §8] or [23, §3]), for each component H of the Hilbert scheme of hypersurfaces in X, and each system of multiplicities m, the sets of points (q1 , . . . , qr ) such that there is Y ∈ H with multiplicity mi at qi form a Zariski-closed subset of Xr . Thus, it will be enough to prove that, given H and m, the existence of Y ∈ H with multiplicity mk at qk for general qk ∈ Cpk implies r n−1 mk . Y · Ln−1 εn−1 (L, q)εn−1 OPn (1), p1 , . . . , pr k=1
So, fix H and m, and assume that for general points qk ∈ Cpk there is a hypersurface Y ∈ H going through qk with multiplicity at least mk . In the local ring of Cpk at ∆(q) = (q, . . . , q), the 1 ⊗ · · ·⊗ x¯pk ⊗ · · · ⊗ 1, k = 1, . . . , r, form a regular system of parameters; abusing notation we call them simply xpk . Let Γ ⊂ Cpk be the irreducible curve defined locally by the equations xp1 = xpi , i = 2, . . . , r, which is obviously smooth at ∆(q), and admits x0 := x¯ p1 ∈ OΓ,∆(q) as a local parameter.
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For every pk = [ξk,0 : · · · : ξk,n ], the ideal Jpk ⊂ OΓ ×X,∆(q) generated by the 2 × 2 minors of the matrix
ξk,0 . . . ξk,n , x0 . . . xn defines the germ of a curve Cp k (the preimage of a line in An+1 ∼ = T∆(q) (Γ × X)) whose projection to X is exactly Cpk ; more precisely, the fiber of Cp k over γ = (q1 , . . . , qn ) ∈ Γ is qk ∈ Cpk . Consider now the diagonals ∆i,j = {(q1 , . . . , qr ) ∈ Cpk | qi = qj }, ∆ = ∆i,j , and U = Γ \ ∆. As the points p1 , . . . , pr are distinct, U is a nonempty open subset of Γ . The assumption on the existence of Y tells us that there is an effective Weil divisor YU ⊂ U × X flat over U whose fiber over (q1 , . . . , qr ) ∈ U belongs to H and has multiplicity at least mi at qi . By the smoothness of Γ at ∆(q), YU can be extended to a flat family Y ⊂ (U ∪ {∆(q)}) × X, and then the condition on the multiplicity of the fibers of Y means that Y contains the arrangement of multiple curves whose germ at ∆(q) is defined by the ideal J=
r
Jpmkk ,
k=1
and this implies that the fiber Yq of Y over ∆(q) has multiplicity at least equal to the order of this arrangement at ∆(q), i.e., at least equal to the maximal integer α such that J ⊂ (x0 , . . . , xn )α . This can be computed equivalently as the order of the completion Jˆ A1 ×X,(0,q) ∼ in O = k[[x]] but, by construction of the curves Cp k , one has Jˆ = Iˆ as defined above, so this order is exactly αm (p1 , . . . , pr ). We remark also that, by the smoothness of X in a neighborhood of q, Y is defined by a principal ideal at q, so we get a Weil divisor Yq ⊆ Yq with Ln−1 · Yq = Ln−1 · Y, multq Yq αm (p1 , . . . , pr ), which together with the definition of the 1-point Seshadri constant, gives Ln−1 · Y εn−1 (X, q)αm (p1 , . . . , pr ), and then it is enough to apply the bound of Remark 11. ✷ Proof of Corollary 5. By Theorem 3 it is enough to see that √ lim εn−1 OPn (1), r n r 1 r→∞
(which in fact means that one has an equality, the converse inequality being well-known). More precisely, we shall prove that given k > 0 there exists sk = sk (n) such that if r sk then for all m = (m1 , . . . , mr ) and general points p1 , . . . , pr , mi k + 1 . αm (p1 , . . . , pr ) √ · n n−1 k+n r
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So let F be a homogeneous polynomial defining a hypersurface of degree d in Pn which has multiplicity mi at pi for general points p1 , . . . , pr . Then, by the genericity of the points, for every permutation σ ∈ Sr there is a polynomial Fσ which has multiplicity mi at the point pσ (i) . Therefore G = σ ∈Sr Fσ is a polynomial of degree D = r! d which has (the same) multiplicity M = (r − 1)! mi at p1 , . . . , pr , and Gk+n has degree (k + n)D and multiplicity (k + n)M at each point. By [24, Theorem 1.1(a)], applied to the ideal I of the (reduced) scheme {p1 , . . . , pr }, this implies the existence of hypersurfaces of degree t (k + n)D/M with multiplicity at least k + 1 at p1 , . . . , pr . Now write m = (k + 1, . . . , k + 1); by√[12, Corollary 1.2], there is sk (n) such that if r sk (n) then αm (p1 , . . . , pr ) (k +√1) n r (again, because the points are general). Therefore we get (k + n)D/M (k + 1) n r and 1 D k+1 d √ , = · n n−1 rM k +n mi r as desired. ✷
Acknowledgments We thank B. Harbourne and S. Kleiman for many valuable comments, which largely improved the paper.
References [1] T. Szemberg, Global and local positivity of line bundles, Habilitation, 2001, 42 pp. [2] J. Demailly, Singular Hermitian metrics on positive line bundles, in: K. Hulek, et al. (Eds.), Complex Algebraic Varieties (Bayreuth, 1990), in: Lecture Notes in Math., vol. 1507, Springer, 1992, pp. 87–104. [3] M. Nagata, On the fourteenth problem of Hilbert, Amer. J. Math. 81 (1959) 766–772. [4] A. Steffens, Remarks on Seshadri constants, Math. Z. 227 (3) (1998) 505–510. [5] G. Xu, Ample line bundles on smooth surfaces, J. Reine Angew. Math. 469 (1995) 199–209. [6] R. Lazarsfeld, Lectures on linear series, in: J. Kollár (Ed.), Complex Algebraic Geometry (Park City, UT, 1993), no. 3, in: IAS/Park City Math. Series, Amer. Math. Soc., Providence, RI, 1997, pp. 161–219, with the assistance of Guillermo Fernández del Busto. [7] O. Küchle, Multiple point Seshadri constants and the dimension of adjoint linear series, Ann. Inst. Fourier 46 (1) (1996) 63–71. [8] P. Biran, Constructing new ample divisors out of old ones, J. Reine Angew. Math. 98 (1) (1998) 113–136. [9] T. Bauer, Seshadri constants on algebraic surfaces, Math. Ann. 313 (3) (1999) 547–583. [10] T. Szemberg, H. Tutaj-Gasi´nska, General blow-ups of the projective plane, Proc. Amer. Math. Soc. 130 (9) (2002) 2515–2524. [11] B. Harbourne, Seshadri constants and very ample divisors on algebraic surfaces, J. Reine Angew. Math. 559 (2003) 115–122. [12] J. Alexander, A. Hirschowitz, An asymptotic vanishing theorem for generic unions of multiple points, Invent. Math. 140 (2000) 303–325. [13] G.V. Choodnovsky, Sur la construction de Rees et Nagata pour le 14e problème de Hilbert, C. R. Acad. Sci. Paris 286 (23) (1978) 1133–1135. [14] J.-M. Hwang, J.-H. Keum, Seshadri-exceptional foliations, Math. Ann. 325 (2) (2003) 287–297.
J. Roé / Journal of Algebra 274 (2004) 643–651
651
[15] A. Iarrobino, Inverse system of a symbolic power, III: Thin algebras and fat points, Compositio Math. 108 (3) (1997) 319–356. [16] H. Tutaj-Gasi´nska, A bound for Seshadri constants on P2 , Math. Nachr. 257 (2003) 108–116. [17] B. Harbourne, On Nagata’s conjecture, J. Algebra 236 (2001) 692–702. [18] L. Évain, Calculs des dimensions de systèmes linéaires de courbes planes par collisions de gros points, C. R. Acad. Sci. Paris 325 (I) (1997) 1305–1308. [19] E. Shustin, Gluing of singular and critical points, Topology 37 (1998) 195–217. [20] T. Keilen, I. Tyomkin, Existence of curves with prescribed topological singularities, Trans. Amer. Math. Soc. 354 (5) (2002) 1827–1860. [21] D. Mumford, The Red Book of Varieties and Schemes, expanded edition, includes the Michigan Lectures (1974) on Curves and Their Jacobians with contributions by Enrico Arbarello, in: Lecture Notes in Math., vol. 1358, Springer-Verlag, Berlin, 1979. [22] M. Lejeune-Jalabert, B. Teissier, Contribution à l’étude des singularités du point de vue du polygone de Newton, PhD thesis, Université de Paris VII, Paris, 1973. [23] J. Lipman, Equimultiplicity, reduction, and blowing up, in: Commutative Algebra (Fairfax, VA, 1979), in: Lecture Notes in Pure and Appl. Math., vol. 68, Dekker, New York, 1982, pp. 111–147. [24] M. Hochster, C. Huneke, Comparison of symbolic and ordinary powers of ideals, Invent. Math. 147 (2) (2002) 349–369.